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Projects and publications of the Applied Mathematics ... · NATIONALBUREAUOFSTANDARDSREPORT 10343 PROJECTSandPUBLICATIONS ofthe APPLIEDMATHEMATICSDIVISION ASemi-AnnualReport JulythroughDecember1969

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    of the


    A Semi-Annual Report

    July through December 1969



  • D













    NATIONAL BUREAU OFfor use within the Governmei

    and review. For this reason

    whole or in part, is net eu

    Bureau of Standards, vVashi

    the Report ha: been specific

    Approved for public release by theDirector of the National Institute ofStandards and Technology (NIST)

    on October 9, 2015.

    )gress accour.ting documents intended

    it is subjected to additional evaluation

    tore listing of this Report, (jither in

    the Office of the Director,

    or, by the Government agency for which

    tal copies for its ov/ii use.




    Conduct research and provide consulting services to the Bureau and other Federal agenciesJn various fields of mathematics important to phy.sical an.d engineering sciences, automaticdata processing, and operations research with emphasis on statistical, numerical andcomhinatorial analysis, and mathematical physics. Develop tools for mathematical work suchas mathematical tables, handbooks, manuai.s, mathematical models and computational methods,and advise on their use. Provide training in disciplines related to these functions.

    NUMI’.RICAL ANALYSIS SECTION: The advancement of computation and the theoryof munerical analysis, particularly in the development of computing algi,rithmapproximations to functions, end methods to facilitate the use of high sp'-^-delectronic computers by subject matter specialists. Design of mathematicaltables; exploratory calculations on automatic machines. Consulting servicesand training, and preparation of manuals in these fields. Research inunderlying branches of pure and applied mathematics, such as matjix algebra,combinatorial analysis, and number theory.

    OPERATIONS RESEARCH SECTION; Development and application of mathematical andcomputational techniques for the analysis, improvement or optimization >fcomplex systems or activity-patterns. This includes (l) research in specificrelevant areas of mathematics, such as linear programming, the theory of linegraphs, and the theory of strategic contests, (2) investigations in the artof constructing useful mathematical models of complex systems, and ofobtaining information about the system by applying analytic or simulationmethods, and ( 3 ) application of these techniques to selected problems,of general methodological significance, arising in the work of the Bureauor of other Government agencies lacking specialized pei'sonncl in this field.

    STATISTICAL EJCINEERIMG SECTION: Consulting services in the applicationof mathematical statistics to physical science expei'iments and engineeringtests, particularly in the design of experiments and in the analysis andinterpretation of data. Research on pertinent topics in probabilityand mathematical statistics. Preparation of reports, manuals, tables,studies of computational methods and other aids to the application efmodern statistical methods .

    SYSTEI-IS DYNAMICS SECTION; Research and consulting in applied mathematicsbasic to physics and engineering, with emphasis on analysis of the dynamicbehavior of complex physical systems. This involves, primarily, thedevelop.ment and application of techniques foi’ solving linear and non-linear system.s of differential equations and integral equations, orcombinations of both. Of concern also is simulation of the behavior ;fphysical systems by means of electronic computers using approximationtechniques and semi-analytic methods. Attention is given to problems inplasma dynamics and the behavior of solid matter and multi compone.'nt, with emphasis on developing matliematical methods of wide rangeof applicability beyond the scope of the immediate problem. Invcstig.atl nisare carried out on the special functions encountered in the analysis and

    algpritliras for their evaluation are prepared.


  • Contents

    Status of Pi’ojects° as of Becembei' 31,1969 ^

    1. Numerical analysis 1

    2. Operations research ‘i-

    3- Probability and mathematical statistics 8

    4. Statistical engineering services 10

    5 . Systems dynamics

    Lectures and technical meetings 1?

    Publications activities 19

    °0nly unclassified material is included in this report.



    Jjly 1, 1S69 through Dacernber 31, 1969


    Marvin L» Minsky, Massachusetts Institute of Technology

    Carl A, Bennett, Battelle Hsmorial Institute Murray S. Klamkin, Ford Scientific LaboratoryCharles R. DsPrima, California Inst, of Technology M. Ho Martin, University of MarylandH, Oo Hartley, Texas A & M University John Riordan, The Rockefeller University

    John Todd, California Institute of Technology


    Edward W, Cannon, Ph,D,, Chief

    Catherine Hartsfleld, Secretary Molly F, Hevonor, Admin. Officer, B.S."Yates S. Bladen, Admin, Officer®

    Russell A. Kirsch, M.S. A.nne A. Holston, B.A. Irene A. Stegun, M.A.

    Margaret L. Dean Gertrude V, Thompson Ruth Zuclcer, B.A.Ida I. Rhodes, M.A. , Consultant

    NlRklERICAJ. ANALYSIS SECTION, Morris Newman, Ph,D., Chief

    Doris M, Burrell, Secretary Joseph Lehner, PhtD,"*"

    Karl Goldberg, Ph.D, Russell L. Merris, Ph.D.Seymour )!aber, Ph.D.

    Frank W, o. Olver, D.Sc.Stephen J. Pierce, Ph.D,Justin C. Schaffert*

    STATISTICAL ENGINEERING lABQPJlT'OPY, Joan R, Rosenblatt, Ph.D,, ChiefHslou K, liu, PhcD., Ase‘t Chief

    Raj Cliandra Bose, Pli.D,, ConsultantShirley G, Bremer, B.A.>'''Eleanor S, BrownVeronica Connor, SecretaryJames J. Filliben, Ph.D.

    David Hogben, Ph.D,

    Brian L, Joiner, Ph.D.Mary G, Natrella, B,A,>!‘

    Sally T. Peavy, B.A.'^^

    SYSTEMS DYNAMICS SECTION, Hans J. Oser, Ph.D,, Chief

    Jeffrey T. Fong, Ph.D.Regina Kopec, Secretary

    S, Richard Kraft, Ph.D,Daniel W. Lozlei', M.Ai,

    Malcolm S. Scott, M.S.’’'Janace , Speckman

    , M.S.Ruth N. Varner, B.A.*Roy H. Wampler, M.A.

    William H. Youden, Ph.D.^+*

    Zol]a G, Ruthberg, M.S.’f'

    Walter L. Sadowskl, Ph.D.

    OPERATIONS RESEARCH SECTION, Alan J. Goldm.ati, Ph.D., Chief

    Jack Edmonds, M,A,°

    Judith F. Gilsinn, M.S.William C. Hall, B.A.Dolores F, Harrison, SecretaryWilliam A. Horn, Ph.D,

    liSmbort S. Joel, B.A,

    David Klavan, B.A.’i’

    David S, Lfibovitz, B.A, “S'

    Jeffery C. Lagarlns

    Joel l,0vy, M.A.

    Pliilip Meyers, Ph.D.t

    Martin H. Pearl, Ph.D.*Patsy L, Saunders, B.S.Roger D, Traub, B.A,*

    *Part-t imo^^Postdoctoral Research Associate***Guest Worker

    °On Leave of Absence^E.O.D, November 5, 1969^Retired October 31, 1969

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  • Status oi Projects



    Task 20501-12-2050110/55-55


    Origin: NBSManager: Morris NevananFull task description: July - September 1954 issue, p. 1

    Authorized 8/29/54

    Status: CONTINUED. M. Nev.mian has proved that if 77 is a unit in the cyclotomic number fieldR(C) > where ^ is a primitive pth root of unity and p is a prime > 3 , then Tj is the rrh

    power of another unit only if1

    rj < log N/log(2 cos-^ ) ,

    where N is the maximum of the absolute

    values of 17 and its conjugates. This result was applied to prove that a number of diophantineequations involving cyclotomic units had only trivial solutions.

    M. Newman and L. Greenberg have proved that the density (in a well-defined sense) ofsolvable groups generated by elements of odd order is zero.

    M. Nc^'/man has prepared several versions of his "exact solutions" program using only basicFORTRAN. These can be used immediately on any machine with a word length of 36 or more bits. Somedozen copies of this program have been distributed by request.

    S. Haber completed vjriting a survey of the literature on numerical evaluation of m.ultipleintegrals


    F.W.J. Olver has continued the study of the method of .stationary phase for the asymptoticexpansions of oscillatory iiAtegrals. An improved form of error analysis has been constructed and ibeing tested on illustrative examples.

    K. Goldberg extended his results on similarity for formal power series over fields,substituting a matrix equation for the determinant equation previously found.

    K. Goldberg extended his tables of the coefficients in formal power series for variousoperations including product, iteration, inversion, and the related power series L^(z) defined

    by L^(f(z)) = L^(z) • f'(z) .'

    S. Pierce has determined all multiplicative maps of the 2X2 matrices over a Dedekindring which do not send all singular matrices to 0 .

    S, Pierce has proved that all members of the orthogonal group of a real positive multi-linear form have only eigenvalues of modulus 1 and only linear elementary divisors.


  • Status of Projects

    S, Pierce and K. Merris have shora that associated transformat ioiiw on higher degreesymmetry desses of tensors regarded as representations of the full linear group are alwaysreducible. Investigations for the case of degree one symmetry classes arc continuing.

    S. Pierce has given a short proof using the Hilbert reciprocity law that -] is a simiof tv;o squares in any cyclotoralc field of order m when in is divisible by a prime

    p s 3 or 5 (mod B)

    R. Merris discovered a generalisation, T,of the trace of a matrix. If A and B

    are complex mXn matrice.s, (A,B) “ T(B'''A) is a positive semidefinite hen.nltian (psdh) form .Suppose H = (H^j) is an mnXmn block psdh matrix where is niXm . Then (T(l'^^)) is [)cdh .

    R. Merris and S. Pierce proved the space of mXn matrices with the generalized traceform is regular on].y if T is the trace. Research in this area is continuing.

    S, Pierce and R. Merris proved that if A is a complex matrix of sufficiently largerank and if all the elementary divisors of K(A) are linear then all the elementary divisors ofA are linear. Here K(A) is an as.soclated transformation based on a group and an irreduciblecharacter of arbitrary degree.

    R. Merris and U. Watkins discovered some conditions forcing isomorphism between symmetryclasses of tensors.

    R. Merris discovered an inequality for generalized matrix functions and block herndtianmatrices


    Publications :

    (1) Stochastic quadrature formulas, S. Haber. To appear in Math. Comp.

    (2) Sequences of numbers that are approximately completely equidistrlbuted . S. Halier. Submittedto a technical journal,

    (3) On the sum £< C n > ^ and numerical integration. S. Haber and C.F, Osgood. To appear inPacific Jour, of Math.

    (4) Numerical evaluation of multiple integrals. S. Haber, Submitted to a technical- journal,

    (5) Lectures on modular forms. J. Lehner. IiMS 61 of the NBS


    (6) AuLomorphic Integrals vjith preassigned periods. .1, Lehne.r , J. of Re.^carch NBS, 73.B, 153-161



    (7) On the multipliers of the Dedekind modular function. J. Lehner. J. of Research NBS, 72B , No. 4,253-261 (1968) ,

    (5) Subgroups oi SL(t,z). M, Newman. J. of Research N8S, 73B, 143-146(1969).

    (9) Some results on unitary matrix groups. M. Newmtm and M. Marcus. To appear in J. of Linear



    (10) Some results on roots of unity, with an application to a diophantlne problem, M, Newman.

    Aequationes Hath. 2, 163-166 (1969).


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  • (11) Isometric circles of congruence groups, M. Ncx'nnnn, Amer. ,1. Matn. 91, 648-656 (1969).

    (12) Principal ideals in matrix rings (v;ith S. Pierce). M. Mewman. J. of Research tlBS, 733,211-213 (1969).

    (13) Normal subgroups of the modular group. L. Greenberg and 11. liewman. Submitted to a technicaljournal .

    (14) A diophantine equation, M. Kevtman. J. London Math. Soc. , 43, 105-107(1968).

    (15) A table of the first factor for prime cyclotomic fields. M. Nevroan. To appear in Math, ofComputation


    (16) Some results on solvable groups. L. Greenberg and M. Newman. To appear in Arch.der Math.

    (17) On Riemann surfaces with automorphism groups. J. Lehner and M. Newman. GlasgowMathematical Journal, Vol. 8, Part 2, 102-112 (1967).

    (18) An enumeration problem for a congruence equation. M. Ne’-tman and R. Brualdi. To appear inJ. of Research NBS


    (19) Why steepest de.scents? F.W.J, Olver, To appear in Proceedings of 1969 SIAM National Meeting,June 10-12, 1969.

    (20) Bounded matrix groups. M. Net'/mari and S. Pierce. To appear in Journal of Linear Algebra.

    (21) Orthogonal groups of positive definite multilinear functionals. S. Pierce. To appear inPacific Jour, of Math.

    (22) Multiplicative maps of matrix semigroups over Dedeklnd rings. S. Pierce. Submitted to atechnical journal.

    (23) Orthogonal decompositions of tensors spaces. S. Pierce. To appear in J. of Research NBS.

    (24) Trace functions I . R. Kerris. Submitted to a technical journal.

    (25) Elementary divisors of higher degree associated transformations. R. Merris and S. Pierce.Submitted to a technical journal.

    (26) A class of representations of the full linear group. R, Merris and S. Pierce. Submitted toa technical journal.

    (27) Isomorphic symmetry classes of tensors. R. Merris and W. Watkins. Submitted to a technicaljournal .


    (28) Partitioned herraitian matrices. R. Merris. To appear in J. of Re.searc,h NBS.




    Task 205-12-2050151

    Origin and Sponsor: NBS 11570 Authori:^ed 12/30/60Manager: A.J. GoldmanFull task description: October-December 1960 issue, p.3

    Status: CONTINUED.

    (1) Demand for miscellaneous consulting and advisory services remained heavy. Section staffprovided such services in 99 recorded instances, 96 involving assistance to NBS staff. The 99ins(;ances totalled to 695 recorded man-hours. Cither agencies assisted included the Civil ServiceCoiirmission, National Science Foundation, Office of Emergency Planning, Maritime Administration,Law Enforcement Assistance Agency, and Bureau of the Budget. Requests from universities, industry,professional groups and journals v;ere also met.

    (2) J. Gllsinn continued collaboration with the Information Processing Technology Divisionin a study, for the Federal Aviation Authority, to improve air traffic designation procedures. Apreviously-X";ritten simulation program was documented and applied to test an initial scheme forassigning radar beacon codes to airplane flights; the results were analyzed and documented. (Reportedhere for convenience; supported under Project 6505453.)

    (3) A.J. Goldman, W.A. Horn, J. Levy and M.H. Pearl completed a collaborative effort with theTechnical Analysis Division on another study for the FAA, this one to analy.ze a proposed nevj"capacity" concept for an airport runway and the associated final-approach airspace. For a singlestream of customers, the concept w’as shown (using tl'.e theory of Markov renev.iai process) to be re-presentable by a simple formula useful for planning and analysis purposes. The theoretical develop-ment and illustrative calculations were documented, and also presented orally at several briefingsfor interested FAA personnel. (Reported here for convenience; supported under Project 4314449.)

    (4) W.G. Hall continued helping the Army Data Field Systems Command in planning, design andnumerical-analysis aspects of a new tactical artillery-fire control system. Efforts includedcontractor monitoring, and design and testing of a short course in basic computer programming andnumerical analysis for tho Fort Sill Artillery Training School. (Reported here for convenience;supported under Project 6505425.)

    (5) Goldman, Hall, Horn, and L.S. Joel continued assistance to the Post Office Department'sBureau of Research and Engineering in technical Interfacing v;ith contractors concerning computerized

    models and data analysis. P. Saunders began assistance to the Technical Anaiysl.s Division in asimulation study of Coast Guard search and rescue operations. (Reported here for convenience;

    supported under Project 4314561.)

    (6) A.J. Goldman began service as Associate Editor of tlie Operations Researcli Society's"Transportation Science" journal, and Vice-Chairman of the Society's Transportation Science Section.


    (1) M. Aronoff (Dlv. 43]), A.J. Goldman al^. Analysis of a capacity concept for runway and

    final-approach path airspace. NBS Report 10 111, 11/69.

    (2) R.D. Elbourn (Div. 650) and J.F. Gilsinn. Sin:ulation of Air Traffic Control radar beacon code

    assignment plans (final report on Phase 1). NBS Report 10 117.

    (3) A.J. Goldman and P.R. Meyers. Slmultane ous contractif ication. Journal of Research NBS,

    73B (1969), pp. 301-305.

    (4) A.J. Goldman. Systems analysis and urban problems (Discussion). To appear in Prcc. Sjnnposium

    on Systems Analysis for Social Problems, 5/69,

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  • W.A. Horn. Some fixed point theorems for compact maps and flows In Banach spaces,in Trans. Ainer. Hath. Soc.

    To appearC3)

    (6) W.A. Horn

    (7). W.A. Horn

    Convex homotopy. Submitted to a technical journal.

    Optimal design of sorting networks. NBS Report 10 146 (1/70).


    Task 205-12-2050152

    Origin and Sponsor: NBS 11540 Authorized 12/10/60Manager: A.J. GoldmanFull task description: October-December 1964 issue, p.3; April-Jur.e 1962 , p.l5

    Status: CONTINUED

    (1) J. Gilsinn (and C. Witzgall of Boeing Scientific Research Labs) continued experimentsand documentation on the comparison of shor test-path algorithms.

    (2) A.J. Goldman continued studies relating to the optimal location of facilities in networks.He and C. Witzgall generalized the theorem that a single node, which produces half or more of thetotal flow, must itself be an optimal location.

    (3) L.S. Joel initiated computer implementation of J. Edmonds' algorithm for minimum-cardinalitypartition of a matroid into independent sets. The program "frame" for general matroids, and thesubroutine to make it applicable to graphical matroids, were programmed.


    (1) J. Edmonds and D.R. Fulkerson (RAND). Bottleneck extrema. To appear in J. Combinatorial Theory.

    (2) J. Edmonds and 0. Shisha (Wright-Pat terson A.F.B.). Acute bijections. To appear in J.Combinatorial Theory.

    (3) A.J. Goldman. Optimal locations for centers in a network. Transportation Science 3 (1969),pp. 352-360.


    Task 205-12-2050153Origin and Sponsor: NBS 11540 Authorized 12/30/60

    Manager: W.G. HallFull task description: October-December 1960 issue, p.3

    Status: CONTINUED

    i| (1) W.G. Hall and P. Saunders continued work to improve the capabilities for linear programming

    :|calculations at NBS. Testing has revealed flaws in several proposed programs; particularly hazardous


    are those in which an erroneous solution (or an incorrect "no solution" indicator) arc given v.Mthout


    ! (2) A.J. Goldman continued studying the minimax error selection of incom.pleteiy specified

    ? discrete probability distributions, examining the problem of disaggregating an exactly or


    approximately "given" distribution.


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  • Pub llc.Tt tons

    (1) A.J. Goldman. Minlinay. error selection of a univariate distribution '.»'itli pre.scrjbcdcomponentwise bounds and ranking. Journal of Research HBS, 73B (1969), pp. 225-210.

    (2) A.J. Goldman. Minimax adjustment of a univariate, distribution tc saci.sfy componentwise boundsand/or ranking. Journal of Research KBG, 7 3B (1969), pp. ^3]-239.

    (3) W.A. Horn. A theorem on convex hulls. Journal Research NBS , 73B (1969), pp . 307-308.


    Task 205-12-2058656

    Origin: Technical Analysis Division, NBS 21550Sponsor: Northeast Corridor Transportation Project, Dept, of TransportationManager: P.B. Saunders

    Full task description: This is one of three projects into which our wor.k for the Northeast CorridorTransportetion project has been split. .Its objective Is to cont.inue thedevelopment, computer impl craentetlori and nppl Icution of metj-iods i,o c/Dt;-iri"good" operating rules for transport systems.

    Status: NEW

    (1) P. Saunders continued work on integrating rail scheduling a.lgorichms and demand forecastingmodels. Contractor reports on rail and air-flight scheduling were reviewed.

    (2) D. Klavan programmed and debugged the algorithm developed by G. Nemhauser (Johns Hopkins U.)for jointly optimal scheduling of local and express service along a line of stations. A slightly

    revised version, to liandle two "locals" which compete over some Interior links of the line, wasalso developed.




    (1) D.R. Young (now of Stanford U,). Passenger transportation scheduling. NBS Report 10 069 (6/69).


    Task 205-12-2050657

    Origin: Technical Analysis Division, NBS 21550Sponsor: Northeast Corridor Transportation Project, Dept, of TransportationManager: A.J. Goldman

    Full task description: This is one of three projects into which our work for the NortheastCorridor Transportation Project has been split. Its objective is to develop,

    calibrate and prototype-test improved passenger-demand forecasting models.

    Status: NEW

    (11 L.S. Joel began an analysis of the promise for niodel use of various additional explanatory

    variables describing characteristics of the transport system. He and .1. Lagarias initiatoil an

    investigation of the relationship between intercity passenger trip volumes and telephone-call




  • (2) A.J. Goldman and R. Ku (Division 431) examined tv;o mo’del inodif icaLion:; : one reflecting

    the idea that auto-driver fatigue increases more than linearly -with trip time, the other revising

    the modal-split submodel to incorporate a probabilistic lower limit on acceptable conductance.

    Neither change significantly Improved the fit to the present da(:a set.

    (3) J. Gilsinn, A.J. Goldman and R. Traub revie\;ed contractor reports and related literature.Ku completed a summary ch.apter on the model cui rently used by the Corridor Project; Goldman (withli. Cheslow of DOT) began a more extensive documen tu t ion




    J. Gilsinn. Roles of demand models in the Northeast Corridor Transportation Project.NBS-NDCTP Working Paper No. 12, 7/69.


    Task 205-12-2058458

    Origin: Technical Analysis Division, NBS 21550Sponsor: Northeast Corridor Project, Department of TransportationManager: A. .J. Goldman

    Full task description: This is one of three projects into which our work for the NortheastCorridor Transportation Project lias been split. Its objective is to providemiscellaneous mathematical services and exploratory research for theNortheast Corridor Transportation Project (NECTP)


    Status: NEW

    (1) J. Levy and M.H. Pearl continued developing methods for evaluating feedback vs. non-feedback flow regulation at a merge point in a transport network. A comparison of the performanceof non-feedback, feedback, and "prophesying" control v:as begun.

    (2) VI. A. Korn completed studying an approach to the design of transport networks when

    minimizing the number of access points is of prime concern. He also assisted in review of contractor



    (3) A.J. Goldman assisted in editing the first draft of the NECTP 12/69 Report.


    (1) J. Levy and M.K. Pearl. Feedback regulation of traffic at an_ intersection of two streams.NBS-NECTP Working Paper No. 11 (9/69).

    (2) M. Krakowski (Tulane U.). Adaptive svritching and routing in transport networks. NBS-NECTP

    Working Paper No. 13 (10/69)'.

    (3) W.A. Horn. Optimal Networks Joining N Points in the Plane. Strassenbau und

    Strassenverkehrstechnik, ^ (i969), (Beitrage zur Theorie des Verkehrsf losses) , pp . 161-166.



    Tasic 201j03-]2-20'^0131/63-I2>91150

    Origin: NBS Authorized 10/]/62Manager; Joan Raup RoaenLla.ttPull tasK description: July - December I962

    Status; COlWIl'IUED. Brian L. Joiner and Eleanor S. Brown, in coDlaboration with Bert Levy (HarryDiamond labs.) and N. F. Laubscher (National Research Inst, for Math. Sciences, South A.frica),completed the preparation of a co;nhined author and permuted title index for seven statistics journals,Including articles published since the most recent cumulative subject index to each of the journals.The index V'as premred using a computer program vn’itten by the late V. W. Youden (NBS Center forComputer Sciences and Technology).

    James J. Filliben is completing a study of the nature of the percent point function (ppf)--the inversecumulative distribution function. It is sliown that a nuniber of statistical procedures liave a simplerand more natural expression in terras of the ppf rather than in terms of the cumulative distributionfunction. Various simplifications in probability theory and order statistic theory are presented.The use of the ppf in random number generation and probability plots is discussed. An alternativeset of measures of distributional characteristics based on the ppf, rather than on the mom.ents, isIntroduced. The heliavior of the ppf under various tj'pes of transformations is considered. The useof the ppf in generating distributions with pre -.'specified tail lengtli is demonstrated.

    Publi cat ions:

    (1) The median significance level and other small sample measures of test efficacy. B. L. Joiner.J. Amer. Statist. Assoc., ^ (1969), 9T1-985-

    (2) Student-t deviate corresponding to a given normal deviate. B. L. JoinerJ. Res. MBS - C. Engineering and Instrumentation, 75C (1969)? 15-16.

    (3) Some properties of the range in samples from Tukey's symmetric lambda distributions.Brian L. Joiner and Joan R. Rosenblatt. Submitted to a technical journal.

    (^) An author and permuted title index to selected statistical journals. Brian I,. Joiner,N. F. Laubscher (Nat'l. Res. Inst, for Math. Sciences, S. Africa), Eleanor S. Brown, andBert Levy (Harry Diamond Labs., IJ. S. Amy). To appear as NBS Special Pub, 321.

    (5) Approximating discrete probability distributions. H. H. Ku and S. Kullback. (G.H.'j.).IEEE Transactions on Information Theory, IT-1 5 (19d9)>

    (6) Analysis of multi -dimensional contingency tables. H. H. Ku, R, N. Varner and S. Kullback(George Washington Unlv.). To appear in Free. l4tli Conf. on Design of Experiments in ArmyResearch Development and Testing. ,

    (7) An application of minimum discrimination information estimatiojn to a problem of Grizzle andBei'Kson on the test of "no interaction" lOTothesis. H. H. Ku and S. Kullback (G.W.U.).Submitted to a technical journal.

    (8) Symmetry and marginal homogeneity of an r x r contingency' table. C. T. Ireland, S. Kullbaci,(G.V’.LI.), aivi H. H. Ku. To appear in J, Araer. Statist. Assoc,

    (9) An evaluation of linear least square.? computer programs, Roy H. Wampler.

    J. Res. KBS-B. Math. Sciences, 73B (1969), 59-90*

    (10) An evaluation of linear least squares computer programs: A sumuary report. Roy il. Wampler.

    To appear in Proc. l4th Conf. on Design cf Experiments in Army Researcli Development and Testing.

    (11) A report on tlie accuracy of some widely used least squares computer programs. Roy H. V/ampler.

    To appear in J. Amar. Assoc.


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    Task 20503-12-205013111550

    Origin and Sponsor: KBS Authorized II/1/68Managers: David Hogben, Sally T. PcavyPull task description: July - December I968

    Status; CONTIIiUED. David Hogben, Sa]jy T. Peavy, Ruth N, Varner and Shirley G. Bremer continuedtrork on the development of the OMNITAB system. M. Stuart Scott developed a comiriand forcorrelation analysis detailed autonatic printing. Fight new versions have been implementedsuccessively. Details of the changes made appeared in the t'nree Newsletters issued. A set oftvrelve instructions for perforniing operations on magnetic tape vrare added to the system.R. C. McClennon (Office of Standard Reference Data-) contributed two conunands for array operations.Other commands were added for printing data, and Gauss quadrature.

    Many improvements were made in the printing and curve fitting commands. Results are now' normallyprinted according to the "readable" format. The automatic printing for the least squares in-structions has many valuabi.e additions, in particular, one page containing four different plotsof standardized residuals.

    An important change W'as made in the word length of alpha/numeric data so that ONiNITAB is now' asmachine independent as the state of the art will pennit.


    (1) The use of OMNITAB for statistical analysis of tabular data. Joan R. Rosenblatt,Brian L. Joiner, and David Hogben. To appear in Census Tract Papers: Papers Presentedat the Conference on Small-Area Statistics, Aunerican Statistical A.ssociation, New York,August 21, 1969

  • 4 n *}

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    Task 13911-612050950/5.1-J.99500

    Origin; NBS Authorised 7/1/5OMauagei’S; H. il. ?ai. J. R, RosenblattFull task description: July - September 1950 issue, p. 60

    Status: COMTINIIED. This is a continuing project involving cooperaticn VTitli other Bureau scientistson the statistical aspects of their investigations. These services vary from short (one-hour)sessions to extended collaborations involving several ina.n-monthsj and are concerned prlmariDy withstetistical design of experiments, analysis and interpretation of data, and the use of cornpiuters instatistical analysis of data. Typical examples of the services performed are the following.

    B;riaa L. Joiner is on detail to the Boulder Jjaboratories for one year, at the request of the NBSOffice of Measurement Services. His primary mission is to heip them establish cooperative calibrationprogratas and to provide better inforination to support the uncertainty statements on the reports ofcalibration. So far effort lias concentrated on the analysis of existing data and the planning ofexperiments to obtain f'urtlier data on oo8,xlal and microvfave bolometers.

    Brian I,. Joiner presented a talk on "Error Analysis" ac a Iteser Povrer and Energy Measurement Seminarsponsored by the Quantum Electronics Section of the Radio Standards Physics Division.

    Brian L. Joiner assisted Johi: Dean and Douglas Mann (Cryogenic Metrology Section) in the planning andanalysis of experiments to evaluate cryogenic flow meters.

    Th.r Statistical Engineering Laboratory continues to distribute within the Bureau occasional notes onuseful subroutines and documentation aids, for computer users who are not primarily computer•pecialists. Three recent notes, for example, describe and give details for using plotting sub-rcjutines and programs applicable with equijjmerit available at the Bureau, Sally T. Peavy andPvUth N. Varner are the chief contributors to the series, but notes have also been ’.nritten by other(ijembers of the Statistical Engineering Laboratory and by persons in other Btireau divisions.

    Roy K. V/ampler and Joan R. Rosenblatt are collaborating with G. L. Hov.'ett (NBS Office of Color: ruetry)on the analysis of the latest in a series of cooperative paired comparison experiments conducted bythe Committee oi: Uniform Color Scales of the Optical Society of America.

    Joan R. Rosenblatt collaborated v.'ith M. A. Bond, Jr. (NBS Library) to develop and execute a simplesam'ole design to estimate the distribution by publ.icat 1 on date of the library's books (not includingjournals)


    G. llatrella presented three lectures on "Statistics of Measurement" for senior tecimicians fromihe State Weights and Measures Departments of Havra.ii, Ca li.foriiia, and TiUnois, Staff meiiibers of the;iBS Office of 'Weights and Measures also attended tlie lectures.

    lar.ies J. Fllliben collaborated with George Hicho (312.0j)and Harvey Yal.owltz (312,03) in the

    pneT'aration and testing design of specimens for X-ray diffraction standards -- about 1000 pellets

    compected from 5 vreight percent 310 stainless steel powder and 95 weight percent 430 stainless steel




    (l) Iriteriaboi-atory comparison of the potential heat test methed, D. Gross (Fire Research Section)

    and M. G. Natrella. Submitted to a technical journal.


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    Task 20503-^0-2050132/58-3^611550

    Origin and Sponsors; Various Agencies Authorized 3/31/58Manager: J. R. RosenblattFull task description: January - March 1958 issue, p. h’y

    Status: CONTINUED. This is a continuing project which involves providing, upon request, statistica.lservices to other governinental agencies, universities, industrial organizations, and otlier non-governmental agencies. Approximately jO such requests are lie.ndled per month langing frcm shortconferences to coi.laboration involving several da,ys vorK,

    Janace A. Speckraan used OMIITAB to perform l44 least soiiares straight line fits to data on theFederal Court of A.ppeals workload, in response to a request from James A. McCafferty of the Divisionof Procedural Studies and Statistics, Administrative Office of the United States Courts.Projections for 1970 to 1975 were computed and plotted for all 11 circuits combined and for eachcircuit separately for 6 variables -- (total nuniber of appeals, numbei* of prisoner petitions, etc.)using data from i960 to 1969. Projections vrere also made using only the data from I965 to I969 tosee if the trends have changed.

    The Statistical Engineering laboratory vras asked by the NBS Center for Computer Sciences andTechnology to participate in one of a series of orientation meetings for a group of foreign trainees(sponsored by the Agency for International Development) from the Census Bureau's Office of Inter-national Statistical Programs. The group included several computer prograinmers with i^articulai’interest in statistical computations.

    In response to requests, twenty-two copies of the OMNITAB program were distributed.


    rese«jk;h lij’ sYS'i'-W'is /rayu-iicf.!


    Taek ?05)i0-12-20i)0l)^l/55-57

    Oi’igin: IJBS Authorli-;cd 9/l/|;)4

    Manager; H. J. OserFull taslc description: July-geptemler 195^i- issue, p. 2"{

    In consultation with Hr. J. A. Siiaaons of the Metallurgy i)iviBion (312-05) and Jlr'. E. A. K'’>i2-cley ofthe Mechanics Di\^slon (213-05), J- T. Fong proved tliai the problem of tlie finite defonaation of amemory-dependent .material continuum of the Bernstein-Kenrsley-Zapas tj^ic-: (Trans. Soc. Rheology, 7,391“^10 ( 1963 ) is equ.ivelent to a variational problem for a rlieonomic system, i.c;. the local Lagrangiondepends on the time t explicitly. A manuscript entitled "On the existence of an action principle fera dissipative continuum" is hoing prepared for review and di.scussJon.

    In collaboration with Mr. L. J. ZapjiS of the Mechanics ilivision ( 213 . 05 ), J. T. Fong showed thot theclassical theory of plasticity in which the time t is not a physical variable but a maihcmiatlcalparameter, can be modeled a.-fter the Bernstein-Kearsl cy-Zapas ' theory cf a memory -dependent materialprovided tlie elastic potential of the EKZ theory Ic a homogeneous fujict ion of the vculable t ' ofdegree -1, where t' is the difference between the current time and some time in the past. Work isC'.ontinviing to derive some basic features of a special pl.a,£;ticity theory from a of the memory-dependent elastic potentials of the bKZ. theory.

    Progress is being made in a long-tcrTu project to find Ih.e necessary end sufficient conditions I'orthe equivalence of the geometric theory of an "imperfect" material continuum due t.o Kondo, Eilby,Kroner, etc. and the history-dependent kineii'.atic theory of a "sljnple" material due to Truesdell, Hell,Kang, etc. To discuss some of his f.lndings, J. T. Fong visited Professor Y. C. Wong of the Mathematics, Uni.versixy of Horxg Kong, as a guest worker with the title of Vlslcing Research Fellowfrom July 5 to July 12, I.969 during which J. T, Fong also delivered the following two talhs


    "Microscopic Theories of Engineering Material.s" - University of Hong Kong;Department of Civil Engineering Seminar, July E, I969 ;

    "Applications of Differential Geometry in Continuum Physics" - Universityof Kong Kong: Dept, of Mathematics Seminar, July 9» 1969-

    R. S. Kraft obtained a constructive existence proof for a pure initial-velue prol.lem associated witli a.linear transport equation. A paper cn tlJ.s subjecc i.s b^ing prepa,red. Ideas fci' deriving a-prioriboimds for an approximation scheme associated with a bouirinry-lniTlal problem for a linear transportequation have been obts.ined.

    Tlie -study of a finite-difference scheme for the Kemst-Planck equations is continuing. Tecimlqucs foreDUablishing the positivity and convergence cf the non-lincar luerates are being examined.

    R. S. Kraft discovered also a possibility for generalizing a method of reflecting solutions of elliptic

    systems. The method, if feasible, would allow continuation across an analytic hcuncKu'y whon the solu-tion of the system satisfies very general condi.tions on the boundary.

    H. J. Oser completed, joi.ntly with J. E- McKinney of the Rheology Section (213-05), two manuscripts v,’ith acoustic propagation in viscous and inviscid, heat-conducting media. The authors show

    that certain solutions of the underlying fourth-order i)ai-tial differential equation are not stable, a

    result which has not been I'epcrted heretofore in the ^Litci'abure. In viscous fluids the instability

    does not occur if tlie vises;- Ivy exceeds a certain critical, value wliich depends on the ratio of the

    specific heats, the Prandtl number, and the viscosity.


  • Continuing his effort to provide au :l-Mproved autoicatic nincerlcnl intcgratio:i routine for OilinTAB,D. V;, Lozier produced end chocked out a Fortran program that uaca on adaptive Simijcon procedure. Healso completed and submitted to American University a paper entitled "Frechet Ill.fferential Calculus"as the finaJ. req^ulrement for the MA degree, which vas awarded in August. In another project, he de-veloped an algorit)m, and produced an Iniplementlng Fortran program, to determine v/itli a iiigh degreeof precision the volume of a nearly perfect si^here from experimental data taken by measuring devia-tions from roundnecs of great circles on the sphere. This work was done for P. E. Pontius and H. A.Bovnoan of 213.31 (Mass and Volume Section, Metrology Pivlslon). Several test cases were run to checkout both the algorithm a;id the program. Finally, he provided jnatheraatical e,nd computer programmingsupport for W. L. Sadowski and other staff members of UBS.

    Z. G. Ruthberg compai'ed the efficiencies of several matrix inversion programs and two programs forfinding roots of polynominals . This effort is intended to involve the Section mere deeply in theprogram for algorithm testing and development.

    Z. G. Ruthherg also continued work on the operator [a(x,y) ^ + h(x,y) • A FORTRANprogram is now avaJ.lable that produces the successive powers n of this operator. Tliis new programis fai* more efficient (concerning the storage requirement) and faster than the earlier LISP program.

    Consultations to and cooperative activities with other Bui*eau units continued throughout the period.These involved contacts with the Metrology ilLvision (now a part of the Atomic Physics Mvision), theMechanics Division, the Electronic Technology rivlsicn and the Meat Division. The Mass and VolumeSection of the Metrology Js concerned with determining with high precision the volume ofalmost perfect spheres . A method vas developed with that group to determine the volume correctionsnumerlcal].y from a suitable number of traces along approximate great circles of the pseudo sphere.Tlie results appear to offer a nev; and considerably more economical way of obtaining precise volumes forpui'poses of density and el?.ss determinations.

    Anothei’ effort for the Laser group in the Metrology U’.vision le devoted to checking whether anomalouslight propagation may take in cerbaln laser optical arrangements . There is some speculationthat such might he the case. The investigations are not conclusive so far.

    Work with J. E, McKinney ( 213. 05) was completed on the propagation of sound in viscous and inviscidmedia. A paper dealing with viscous case will appear in Jour, of Acoustical Society of America inthe January 1970 issue. . The paper on the inviscid case was subriiitted to the V/ashington EditorialReview Board for approval to appear in the KBS Journal of Research.

    Shorter consultations with mejnbers of the Heat Division and the Electronic Technology Division involvedthe solution of Integral equations and bo-ondary vad.ue problems


    H. J. Oser sei-vcd as editor and trons]iitor of a book on the theoi-y and computation of hoimdary layers,published by M.I.T. Press Inc.

    Publications :

    (1) Stability of Acoustic Waves witkin a Viscous, Coraprossible. Heat-Conductirjg Fluid. J. E. McKinneyand H. J. Oser. To appear in the Jour, of Acoustical- Society- of America, Vol. 47, Wo. 1. Jan.


    (2) A. V/alz (edi.ted and transljated by H. J. Oser): Boundary Layers of Flow and Temperature. M.I.T.Press, Inc., October 19d9-

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    Task 20504-12-205014o/59-M(-2II5I1O

    Origin: KBS /luthorized 10/].3/6lMejiager: VJalter L. SadovskiFull task description; October-Eeceiater I96I issue, p. 12

    A hlgh-ordor Taylor series expenslon in the time vsxiable for the numerical solution of the Vlasovequation has been developed. The expansion pei'nlts the use of a smaller number of Hemlte poljTiomlnalsand has a very' sma^J. truncation error. The error is expllcltely' calculated. In addition to thehigher speed, the algorithm makes it possible to study truncation errors in both Fourier ar/d Hermltespace


    The results show that the use of the electric field as 0 ci'iterion for numerical stability' is wholly'Inadequate, numerically' unstable runs giving substantially the same value of the electric field asstable runs. This Indicates that nujnerical studies of processes that are highly dependent on kineticeffects in plasma, such as hump on tall, must he re-examined to Insure that the results obtal.ned werenumerically accurate.

    Work is in progress to determine the ridnlmum size of eigenfunction expansion sets for various values ofparameters. A paper on this topic is in preparation.

    A one-day vrorkshop was held at I'lBS on October 15, 19^9 • The 'workshop dealt with the numerical solutionof the Vlasov equation. Amorg the topics discussed were the relative merits of the different methodsof solving the equation, reliability of results, numerical difficulties encountered and ways of dealingwith these.

    Biring this conference the paidiclpaivts decided that a file of preprints should be held at HB3 and anewsletter be sent out to the participants to keep thera Informed on wliat the various groups aroimd the

    country are doing. Ho agreement on Intercomporison of results has been reached, but a t’wo-v.ieek working

    session, possibly at I'fflS, has been suggested for the summer of 1970 and the matter will be taken upagain.


    (1) Temperature Renormalization in the Non-Linear Vlasov Equation. W. L. Sadovski and Z. G. Ruthberg.

    J. of Reseai’ch NBS, Vol. 73B, pp. 28I-29I, Dec. I969.

    (2) Selective E>:citatlon of Harmonics in a CoUisionless Plasma by Two Countorstreajnlng Electron Beams.W. L. Sadovski and Z. G. Ruthberg. J. of Reseta'ch NBS, Vol. 73B, pp. 293-300, Dec. I969.

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    Task 20500-12-2050'i04


    Origin: NBSSponsor: National Institutes of HealthManager: Russell A. KirscliFull task description: January-June 196A issue. 19

    Autliorized 1/21 /6A

    Status: CORTpiUED, The general procedure developed on the Q32 computer for automatic segmentationof quantized images which yields as output a partial ordering (induced by the inclusion relation)of all morphological objects in an image was continued as elements of this partial ordering servedas inputs for a higher level syntactic analysis. These procedures were applied to optical serialsection photographs of a double neuron which v/ere made using a progr'r.mmablc microscope stage andfocus connected to the LINC-FDP-8 computer. The sections v;ere the’.i reconstructed and the morphologicanalyzer was also used to reconstruct an optical intensity photograph of two c’aromosomes



    Task 20500-2050408


    Origin: NIiS Authorized 6/18/68Sponsor: National Institutes of HealthManager: Russell A. KirschFull task description: July-December 1968 issue, p. 14

    Status: CONIINITED. Progress was made in the study of the use of a programmable scanning microscopecontrolled by a PDP-8 digital computer.


    Task 20500-2050410


    Origin: NBS Authorized 6/27/68Sponsor: National Institutes of HealthManager; Russell A. KirscliFull task description: July-December' 1968 issue, p. 14

    Status: CONTINUED. Further efforts v;ere made to coordinate the use of remote computers in communi-cating pharmacological information regarding chemical structure and biological activity among a groupof collaborating pharmacologists. Advisory work continued on Chemiccl Biological Information Handlintechniques



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    Task PO5OO-I1O-205O] 2 .1 /57-216


    Origin end Sponsor; NBSManager: I. A. StegunFull task description: July -Pecern’oer 196)4 issue, p. )4


    Stetus; COHTU'TUED. Consultation continued with the steff of KBS, other government agencies, Industr-yand universities. The topics covered were tables of specie! functions, general computing techniquesend compute tional pitfalls,

    I. A. Stegun end R. Zucker, with emphasis on specie! functions, have conblnued(l ) the file maintenanceof available algorithms, spproximetlons and programs (2) the critical exemlnetlon of availablealgorithms end opiproxlraations (3) the testing of the implementing subroutines. A paper has beenprepared on automatic computing methods for special functions- in particular- the error and com-plementary error function, The FORTRAN program yielding full machine accuracy for both functionsover the entire range will be presented together \;ith the test program and sample results.

    In the course of the testing of subroutines, hardware and software errors have lieen detected, inparticular, double precision mul tiplicetJ.on yielded incorrect single precision results under certainconditions; the compiler set up the computation of an external factor as an internal factor in'a suriimation. In collaboration with the Center foi' Computer Sciences and Technology, the hardwareerror was corrected end the software error is being further analysed.

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  • Some oi' the aspects

    Oav/son, J.

    Armstrong, T,

    Feix, Iih

    Danov i 1 1 , J


    Mere I', H.

    Kruskal, M.

    Symoii, K.

    Sadov,-ski, W„

    Sliero, Kenneth D.

    Gilsinu, David

    Marlin, Monroe H.

    ilOGBEN, D,

    WALKER, J. C.

    OLVER, F, W. ,J.

    SADO’.VSKI, W. L.

    JOEI., L. S.

    Lectures jchnical Mr *' rra

    Workshop on Vlasov Equations

    October 15, 19G9

    discussed were as follows


    (Princeton University) Pi'.rticle v,s. distribution function inodojs

    (University of Kansas) Interconiparison of eigenfunction oxjiansion with theLos Alamos codes

    (University of Nancy, France) Tlio waterbag inodel

    (NRL) Fourier-Fourier transform niothod for the solution of the Vlasov equation

    (Oak Kidge i^aboratory) On some aspects of truncation in Fourier-Fouriortransforms of the Vlasov equation

    (Princeton University) Some remarks on the f;tability

    (Madison, Wisconsin) Bit pushing for simulation of particle models

    (Systems Dynamics Section) Setting up of a coordination center and the problc-;-.of intercomparison of numerical lesults

    Systems Dynamics Seminal'

    NASA-Goddard Space Flight Center and University of Nebraska, Lincoln, Nebraska.Oscillation Theorems for a Class of Delay Equations. July 23.

    Melpar, Inc. (A Division of .American Standai-d Co,), Falls Church, Va . Domainsof Stability of Integral Manifolds for Some Nonlinear Differential Equations.•Nov. 12,

    Institute for Fluid Dynamics, University of !\!arylar.d. A Now Method for FindingExact Solutions of the Navicr-Stokes Equations. Doc. tO.

    Mathomatics Division Expository L.c t nj_e_s

    Computing for the Countless. July 23.

    (Center for Computer Sciences and Teclmology) Categorical Algebra as a New

    Approach to Mathematics. September 17.

    Tlie Method of Chester, Friedman and Urseli Asymptotic E.ypansion oi Definite

    Integi'als Having Two Neai’ly Coincident Sand lo -Poi n ts . October 22.

    The Numerical Solution of Differential Equations uy an Nth-Oixler Ku.nge- Ju t ta

    Method. Deceinber 3.

    Loose Ends in tlie Tlieory of Games. Deterabei' IV.

    NBS Graduate School,Fall .Semester

    PEAtCf, S. T. Fortran Programming for Beginners

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    Dt. 18.

    HOGBEN, David Computing for Die countless. American Statistical Association, New Y'ork,August 19.

    JOINER, Brian L. The use of OMNITAB foi- statistical analysis of tabular data. Amei'icauStatistical Association, Conierence on Small Area Statistics, New Yoi-k,August 21,

    KIRSCM, R. A. Using Computers as Visual rather than verbal devices. D, C. Teachers'College, October 16.

    KIRSCII, R. A. Computer determination of the constituent structure of biological images.Computer Science colloquium, Rutger.s University, December 5.

    KRAFT, R. S. Convergence of semi-discrete approximations of linear tr.ansport equations.University of Oregon, Eugene, Oregon, August 29,

    KU, H. H Analysis of multidimensional contingency tables: An inforruation theoreticapproach. International Statistical Institute, London, September 5.

    NEWMAN, M. Number theoretic subroutines lor high speed digital com])uters, O.xfoi-d,England, August 20.

    Computational problems connected with certain matrix groups, Oxford,England, August 21.

    Normal subgroups of the modular group, Madison, Wisconsin, October 28.

    VARNER, Ruth N. OMNITAB. Washington Statistical Society and Urban Regional InformationSystems Association, Septcml)er 30.




    1.2 Monograph

    Lr;ctures on Modular Forms. J. Lehner, AMS 61.

    l.H Technical Papers

    Minimax Error Selection of a Univariate Distribution witli Prescribed Componentwise Hounds andHanking. A. J. Goldman. J. of Research KBS, 73B 0939), pp. 225-2.:-l0.

    Minimax Adjustnicnt of a Univariate Distribution to Satisfy Cornpohentwiso Bounds and/or Ranking.A. J. Goldman. J. of Research NES, 73B (19G9), pp, 231-239.

    Simultaneous Contractification. A. J. Gol. L. Sadowski and 2, G. HutlibcrJ. of Research NBS, Vol. 7^. No. 4, pp. 281-291 , (Dec . 1969).

    Selective Excitation of Harmonics in a Col 1 is i onless Plasma by Two Counterstrcariing Electron

    Beams. W. L. Sadowski and Z. G. Uutliberg. J. of Research NBS. Vol. 73D , No. 4, pp. 293-300,

    (Dec. 1969).

    An Evaluation of Linear Least Squares Computer Programs. Roy H. Wampler, J. of ResearcliNES, 7^ (1969) 59-90.


  • 1 » 5 Books

    A. Walz (oditocl and IransJatod by li. J. Osor) : BovnuUiry Bayers o.t Flow and Temperature.M.l.T. Press, Inc. Octolier ]969.

    2.0 mNuseniPTs in tup process of publication

    ' 2.3 Technical Papers

    INormal Subgroups of the Modular Group, I.. Greenberg and M. Nev/man, Submitted to a leelmicnj

    I journal,


    Some Results on Solvable Groups. L, Grcenbe)-g and M, Newman. To ajjpear in Arcli. der MaUi./

    Sequences of Numbers that are Approximately Completely Equi dis tribu t cd . S, Haber. To ai)neai-in J. of Assoc, for Computing Machinery.


    Niunerical Evaluation of Multiple Integrals. S. Haljer. Submitted to a tcclinieal jouj-nal .s y -t' On the Sum L and Numerical Integration. S. Haber and C. F. Osgood. To appea:- in

    Pacific J. of Math.

    Symn,ctry and luarginal Homogeneity of an r x r Contir.gency Talilo, C. T. Ireland, S. Kuilba

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  • An I'.v'al ua I ion oX Li.noar Least Squares '’A)ini>u ter Progi'ams : A suiiimaiy ioi>ort. lioy li. IVaiiiplei-.To apjiear in Pjoc . 14th Conf , on Desit;n of Experiments in Army Researcli Development and Testing..

    A Report on the Accuracy of Some Widely Used I

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    iBS TECmiChl Pl5BL[CAT^G^«iS


    JOURNA.L OF RESEARCH reports NationalBureau of Standards research and development inphysics, matliematics, chemistry, and engineering.Compreliensive scientific papers gi\’e conijjlete. detailsof tiie work, including laboratory data, ocperimcntalprocedures, and theoretical and mathematical analy-ses. Illustrated with photographs, drawings, andcharts.

    Published in three sections, available separately:

    @ Physics and Chemistry

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    these fields. This section covers a broad range of

    physical and chemical research, with major emphasison standards of physical measurement, fundamentalconstants, and properties of matter. Issued six timesa year. Annual sub.scription : Domestic, $9.50; for-eign, $11.75*.

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    data processing, and development of test metliods.It will also cover some of the work in acoustics,applied mechanics, building research, and cryogenic

    engineering. Issued cjuarterly. Annual subscription:Domestic, $5.00; foreign, $6.25*.


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    Federal Irsformation Processing Standards Pub-licatiorES, This series is the official publication within

    the Federal Government for information on standardsadopted and promulgated under the Public Law89—306, and Bureau of the Budget Circular A—86entitled. Standardization of Data Elements and Codesin Data Systems.

    Superintendent of DocumentsGovernment Printing OfficeWasliington, D.C. 20402

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